A ratio approach based on the simple ratio test associated with the terms of homotopy series was proposed by the author in the previous publications. It was shown in the latter through various comparative physical models that the ratio approach of identifying the range of the convergence control parameter and also an optimal value for it in the homotopy analysis method is a promising alternative to the classically used h-level curves or to the minimizing the residual (squared) error. A mathematical analysis is targeted here to prove the equivalence of both the ratio approach and the traditional residual approach, especially regarding the root-finding problems via the homotopy analysis method. Examples are provided to further justify this. Moreover, it is conjectured that every nonlinear differential equation can be considered as a root-finding problem by plugging a parameter in it from a physical viewpoint. Two examples from the boundary and initial and value problems are provided to verify this assertion. Hence, besides the advantages as deciphered in the previous publications, the feasibility of the ratio approach over the traditional residual approach is made clearer in this paper.